February 21, 2022

Goals (concepts / buzzwords / jargon)

  • The Population Equation
  • Closed vs. Open Population
  • Unconstrained (density independent) Growth
  • Geometric Growth = Exponential Growth
  • Discrete Time vs. Continuous Time Models
  • Difference Equations vs. Differential Equations
  • Population growth parameters:
    • continuous / discrete / per capita / per unit time
  • Amazing properties of exponential and log.

Meet … Enhydra lutris

sea otter

  • The largest …
  • The smallest …
  • The furriest …
    • guess how may hairs per in2? (hint, humans are born with ~100,000 TOTAL.)

Sea otters: Range

sea otter range

Littorally the entire North Pacific

Sea otters: Keystone Species

sea otter keystone

(Estes et al. 1974)

Sea otters: Furriness > Cuteness

fur trade

  • Fur trade (Russian -> British -> American) leads to near extirpation across the entire range.
  • > 300,000 in 1740 … < 2,000 in 1900.
  • Displacement and indenturing of Indigeneous fishermen (esp. Aleut)

> … the rush for the otters’ “soft gold” was a predictable boom and bust … a cautionary example of unsustainable resource use, and a socioeconomic driver of Western — mainly American — involvement in the Pacific region, (Loshbaugh 2021)

Sea otters: Not totemic

fur trade

art by John Livingston

… a sea otter hunter who was “lazy, spiteful, malicious or disregarded the teaching of the elders” would find his prey “cavorting around his baidarka … teas[ing] and splash[ing] him with water.” Venianimov

  • Ainu - Esaman
  • Koryak - Kalan
  • Aleut - Chngatux
  • Alutiiq - Arhnaq
  • Tlingit - Yáxwchʼ
  • Haida - Ku
  • Nuu chah nulth - Кwak̕aƛ
  • Siletz - Elakha

Sea otters: But culturally significant

Sea otter reintroduction: Pacific NW

Remnant populations from Aleutian Islands … released in OR, WA, BC and SE-AK 1969 – 1972.

Reintroduction trade Reintroduction trade

Sea otter reintroduction: Washington State …

1970: 60 otters

2010’s: over 1000

Successful!

Population ecology is all about …

  • \[\huge N\]
  • but where? when?

Here! Now! …

\[\huge N_t\]

  • but how many were there?

That many, then (\(\Delta t\) ago)!

\[\Large N_t = N_{t - \Delta t} + \Delta N\]

slight rearrangement:

\[\Large N_{t+1} = N_t + \Delta N\] For now, \(\Delta t = 1\), i.e. it’s the discrete unit that we measure population change. VERY TYPICALLY - whether because of biology or field seasons: \[\Delta t = 1\,\, \textrm{year}\].

How does population change?

\(\Large N_{t+1} = N_t + (B - D) + (I - E)\)

Birth

Death

Immigration

Emigration

Assumption 1: no one’s getting on or off the bus

\(\Large N_{t+1} = N_t + B - D\)

Birth

Death

Immigration

Emigration

This is a closed population …

Assumption 2: the important one

The number of Births and Deaths is proportional to N.

\[\Large N_{t+1} = N_t + bN_t - dN_t\] What does that mean?

  • Every female gives birth to the same number of offspring?
  • Every female has the same probability of giving birth?
  • Every female has the same probability of giving birth to the same distribution of offspring?
  • A fixed proportion of all individuals dies?
  • Every individual has the same probability of dying?
  • the distribution of probabilities of dying is constant?

Some math ….

Define \(r_0\) = intrinsic growth, i.e. proportion increase per unit time): \[r_0 = b - d\]

\[N_{t+1} = N_t + r_0 N_t\] \[N_{t+1} = (1 + r_0) N_t\]

\[N_{t+1} = \lambda N_t\]

If \(d > b\), \(\lambda < 1\). If \(b > d\), \(\lambda > 1\).

Cranking this forward

\[N_{t+1} = \lambda(N_t)\] \[N_{t+2} = \lambda(N_{t+1}) = \lambda^2 N_t\] \[N_{t+3} = \lambda^3 N_t\]

Solution:

\[\large N_{t+y} = \lambda^y N_t\] or

\[\huge N_t = \lambda^t N_0\]

Geometric (same as Exponential) growth.

Some examples

Exponential can be very very very fast

Discrete Model to Continuous Model

Let’s do some trickery, starting with: \[N_{t+1} = (1 + r_0) N_t\] \[N_{t+1} - N_t = r_0 N_t\] \[N_{t+\Delta t} - N_t = r_{\Delta t} N_t\]

\[\lim_{\Delta t \to 0} {\Delta N \over \Delta t} = \lim_{\Delta t \to 0} {r_{\Delta t} \over \Delta t} N\]

Magically define: \({r_\Delta \over \Delta t} = r\) and rewrite \(\Delta\) as \(d\):

\[\large {dN \over dt} = r N\]

Solving this differential equation …

\({dN \over dt} = r N\) where \(N(t = 0) = N_0\)

Calculate:

\(\begin{align} {1\over N} dN &= r dt \\ \int_{t' = t_0}^t {1 \over N(t)} dN &= \int_{t' = t_0}^t r dt \\ \log(N) &= rt + C_0 \\ N &= e^{rt + C_0} \\ \end{align}\)

Solution:

Plug in: \[N(0) = N_0\\\] \[ \large N(t) = N_0 e^{rt}\]

Important fact: The exponential function is the ONLY function whose derivative is proportional to itself.

Some examples

Note crazy fast growth

and how straight on a log-scale

Discrete vs. Continuous Modeling

Difference equations \[N_{t+\Delta t} - N_t = \lambda_{\Delta t} N_t\]

think of absolute change

Pros:

  • Reflects (often) biological reproduction patterns, practical sampling schedule (esp. annual)
  • Intuitive

Cons:

  • Depends on discretization timescale
  • Surprisingly difficult to analyze

Differential equations \[{dN \over dt} = r N\]

think of rates (change/time).

Pros

  • Easier “elegant” mathematical analysis
  • Scales nicely

Cons

  • Unbiological
  • Unintuitive

Estimating some rates … discrete

The amazing thing is, if you have an equation “solved”, you only need 2 points on the curve to compute.

Let’s use the discrete equation:

\[N_{t+y} = \lambda^y N_t\] \[1000 = 60 \times \lambda^{40}\] \[16.7 = \lambda^{40}\] \[2.81 = 40 \times \log (\lambda)\] \[0.07025 = \log (\lambda)\] \[\lambda = \exp(0.07025) = 1.0728\]

i.e. population increase about \[7.28\%/year\].

Estimating some rates … continuous

Let’s use the discrete equation:

\[1000 = 60 \times e^{40\,r}\] \[16.7 = e^{40 \, r}\] \[2.81 = 40 \times r\] \[r = 0.07025\] Intrinsic growth rate = 0.07025.

Washington sea otter fit to data

How long does it take for a population to double?

solve for \(t_d\)

Continuous:

\[2N = N e^{rt_d}\] \[\log(2) = rt_d\] \[t_d = \log(2)/r\]

Discrete:

\[2N = N \lambda^{t_d}\] \[\log(2) = \log(\lambda) t_d\] \[t_d = \log(2)/\log(\lambda)\]

Sea otters: ~10 years.

Sea otter references

  • J. A. Estes, J. F. Palmisano. 1974. Sea otters: Their role in structuring nearshore communities. Science 185, 1058–1060.
  • Loshbaugh S. 2021. Sea Otters and the Maritime Fur Trade. In: Davis R.W., Pagano A.M. (eds) Ethology and Behavioral Ecology of Sea Otters and Polar Bears. Ethology and Behavioral Ecology of Marine Mammals.
  • Gilkinson, A.K., Pearson, H.C., Weltz, F. and Davis, R.W., 2007. Photo‐identification of sea otters using nose scars. The Journal of Wildlife Management, 71(6), pp.2045-2051.
  • Veltre, D.W. “Unangax̂: Coastal People of Far Southwestern Alask,”